22:
1334:
891:
1473:
we can consider its induced diffeomorphism on local transversal sections through the endpoints. Within a simply connected chart this diffeomorphism becomes unique and especially canonical between different transversal sections if we go over to the
549:
1478:
of the diffeomorphism around the endpoints. In this way it also becomes independent of the path (between fixed endpoints) within a simply connected chart and is therefore invariant under homotopy.
1123:
782:
787:
770:
338:
376:
1388:
172:
1443:
303:
1580:
theory enters and allows for the geometric interpretation given above. In the case that the extension is already Galois, the associated monodromy group is sometimes called a
274:
1748:"Group-groupoids and monodromy groupoids", O. Mucuk, B. KılıƧarslan, T. ĀøSahan, N. Alemdar, Topology and its Applications 158 (2011) 2034ā2042 doi:10.1016/j.topol.2011.06.048
1471:
1126:
611:
582:
465:
405:
991:
224:
1592:
1345:
it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy classes of) lifts of paths in the base space
933:
1152:
was the first to obtain results towards its resolution. An additive version of the problem about residua of
Fuchsian systems has been formulated and explored by
470:
1153:
1337:
A path in the base has paths in the total space lifting it. Pushing along these paths gives the monodromy action from the fundamental groupoid.
105:
of transformations acting on the data that encodes what happens as we "run round" in one dimension. Lack of monodromy is sometimes called
33: \ {0} gives different answers along different paths. This leads to an infinite cyclic monodromy group and a covering of
1641:
1058:
For a regular (and in particular
Fuchsian) linear system one usually chooses as generators of the monodromy group the operators
1076:
886:{\displaystyle {\begin{aligned}F(z)&=\log(z)\\E&=\{z\in \mathbb {C} \mid \operatorname {Re} (z)>0\}\end{aligned}}}
1073: + 1 when one circumvents the base point clockwise, then the only relation between the generators is the equality
1011:. The covering map is a vertical projection, in a sense collapsing the spiral in the obvious way to get a punctured plane.
1065:
corresponding to loops each of which circumvents just one of the poles of the system counterclockwise. If the indices
1651:
1000:
and the covering space is the universal cover of the punctured complex plane. This cover can be visualized as the
741:
1588:
1787:
1777:
308:
1742:
343:
1342:
97:
as we "run round" a path encircling a singularity. The failure of monodromy can be measured by defining a
1792:
1737:
1352:
1052:
136:
1412:
279:
408:
244:
86:
1782:
1177:
435:
238:
1452:
1797:
1276:
74:
587:
558:
441:
381:
955:
184:
1732:
90:
1032:
1024:
1020:
717:
70:
58:
1759:
1643:
Karl
WeierstraĆ (1815ā1897): Aspekte seines Lebens und Werkes ā Aspects of his Life and Work
1705:
902:
8:
1619:
1503:
1048:
179:
102:
1709:
1683:
1510:
1475:
1258:
653:
618:
66:
62:
21:
1713:
1647:
1614:
1495:
1243:
1225:
1144:
from these classes satisfying the above relation? The problem has been formulated by
1036:
721:
412:
126:
122:
26:
1764:
H. Å»oÅÄ
dek, "The
Monodromy Group", BirkhƤuser Basel 2006; doi: 10.1007/3-7643-7536-1
1697:
1693:
1609:
1262:
1156:. The problem has been considered by other authors for matrix groups other than GL(
713:
1129:
is the following realisation problem: For which tuples of conjugacy classes in GL(
1701:
1577:
1541:
1514:
1269:
997:
42:
544:{\displaystyle p_{*}\left(\pi _{1}\left({\tilde {X}},{\tilde {x}}\right)\right)}
1548:
1185:
1149:
1145:
1771:
1023:, where a single solution may give further linearly independent solutions by
407:. There are theorems which state that this construction gives a well-defined
1398:. The advantage is that we can drop the condition of connectedness of
1581:
1559:
175:
82:
1604:
1031:
in the complex plane have a monodromy group, which (more precisely) is a
50:
1752:
1688:
1406:
1173:
1172:
In the case of a covering map, we look at it as a special case of a
1391:
1333:
1293:
1001:
38:
1027:. Linear differential equations defined in an open, connected set
81:
comes from "running round singly". It is closely associated with
1043:, summarising all the analytic continuations round loops within
89:; the aspect giving rise to monodromy phenomena is that certain
1674:
V. P. Kostov (2004), "The
DeligneāSimpson problem ā a survey",
1014:
1758:
P.J. Higgins, "Categories and groupoids", van
Nostrand (1971)
1287:
to adjacent ones. The effect when applied to loops based at
896:
then analytic continuation anti-clockwise round the circle
555:
if and only if it is represented by the image of a loop in
1047:. The inverse problem, of constructing the equation (with
1257:
In differential geometry, an analogous role is played by
1562:
of the extension is called the monodromy group of
1118:{\displaystyle M_{1}\cdots M_{p+1}=\operatorname {id} }
1069:
are chosen in such a way that they increase from 1 to
750:
1455:
1415:
1405:
Moreover the construction can also be generalized to
1355:
1188:
for simplicity) as they are lifted up into the cover
1079:
958:
905:
785:
744:
590:
561:
473:
444:
384:
346:
311:
282:
247:
187:
139:
1328:
1167:
1224:, and to code this one considers the action of the
1004:(as defined in the helicoid article) restricted to
77:. As the name implies, the fundamental meaning of
1465:
1437:
1382:
1117:
985:
927:
885:
764:
605:
576:
543:
459:
399:
370:
332:
297:
268:
218:
166:
1769:
1646:(in German). Springer-Verlag. pp. 200ā201.
1137:) do there exist irreducible tuples of matrices
1547:This extension is generally not Galois but has
1481:
1279:allows "horizontal" movement from fibers above
776:, but with different values. For example, take
1730:
1639:
1673:
1015:Differential equations in the complex domain
876:
838:
765:{\displaystyle \mathbb {C} \backslash \{0\}}
759:
753:
241:under the covering map, starting at a point
29:. Trying to define the complex logarithm on
1640:Kƶnig, Wolfgang; Sprekels, JĆ¼rgen (2015).
1687:
848:
746:
333:{\displaystyle {\tilde {x}}\cdot \gamma }
1633:
1332:
712:These ideas were first made explicit in
664:. The image of this homomorphism is the
551:, that is, an element fixes a point in
20:
18:Mathematical behavior near singularities
1770:
1297:group of translations of the fiber at
1390:. The result has the structure of a
1192:. If we follow round a loop based at
371:{\displaystyle {\tilde {\gamma }}(1)}
1449:. Then for every path in a leaf of
1180:to "follow" paths on the base space
996:In this case the monodromy group is
378:, which is generally different from
1445:a (possibly singular) foliation of
1383:{\displaystyle p:{\tilde {X}}\to X}
167:{\displaystyle p:{\tilde {X}}\to X}
13:
1458:
1438:{\displaystyle (M,{\mathcal {F}})}
1427:
938:will result in the return, not to
298:{\displaystyle {\tilde {\gamma }}}
14:
1809:
1329:Monodromy groupoid and foliations
1168:Topological and geometric aspects
269:{\displaystyle {\tilde {x}}\in F}
93:we may wish to define fail to be
57:is the study of how objects from
1589:Galois theory of covering spaces
1051:), given a representation, is a
1019:One important application is to
1313:that measures the deviation of
738:of the punctured complex plane
1698:10.1016/j.jalgebra.2004.07.013
1667:
1587:This has connections with the
1466:{\displaystyle {\mathcal {F}}}
1432:
1416:
1374:
1368:
968:
962:
915:
907:
867:
861:
821:
815:
799:
793:
597:
568:
525:
510:
451:
391:
365:
359:
353:
318:
289:
254:
213:
207:
158:
152:
1:
1724:
1582:group of deck transformations
1317:from the product bundle
112:
73:behave as they "run round" a
1482:Definition via Galois theory
1301:; if the structure group of
1216:; it is quite possible that
1200:, which we lift to start at
613:. This action is called the
606:{\displaystyle {\tilde {x}}}
577:{\displaystyle {\tilde {X}}}
460:{\displaystyle {\tilde {x}}}
400:{\displaystyle {\tilde {x}}}
85:and their degeneration into
7:
1738:Encyclopedia of Mathematics
1717:and the references therein.
1598:
986:{\displaystyle F(z)+2\pi i}
772:may be continued back into
702:topological monodromy group
219:{\displaystyle F=p^{-1}(x)}
10:
1814:
1494:) denote the field of the
707:
700:whose image is called the
25:The imaginary part of the
1593:Riemann existence theorem
1178:homotopy lifting property
1626:
720:, a function that is an
305:. Finally, we denote by
1731:V. I. Danilov (2001) ,
1246:on the set of all
1127:DeligneāSimpson problem
1053:RiemannāHilbert problem
668:. There is another map
1753:Topology and Groupoids
1540:) determines a finite
1467:
1439:
1384:
1338:
1309:, it is a subgroup of
1119:
1021:differential equations
987:
929:
887:
766:
617:and the corresponding
607:
578:
545:
461:
401:
372:
334:
299:
270:
220:
168:
46:
37: \ {0} by a
1788:Differential geometry
1778:Mathematical analysis
1622:(of a punctured disk)
1468:
1440:
1385:
1336:
1120:
1049:regular singularities
1033:linear representation
1025:analytic continuation
988:
930:
928:{\displaystyle |z|=1}
888:
767:
718:analytic continuation
608:
579:
546:
462:
402:
373:
335:
300:
271:
221:
169:
71:differential geometry
59:mathematical analysis
24:
1453:
1413:
1394:over the base space
1353:
1343:fundamental groupoid
1208:, we'll end at some
1077:
956:
903:
783:
742:
734:in some open subset
716:. In the process of
588:
559:
471:
442:
382:
344:
309:
280:
245:
185:
137:
1620:Mapping class group
684:) ā Diff(
662:algebraic monodromy
121:be a connected and
1793:Algebraic topology
1558:). The associated
1511:field of fractions
1496:rational functions
1463:
1435:
1380:
1339:
1259:parallel transport
1115:
983:
925:
883:
881:
762:
654:automorphism group
636:) ā Aut(
603:
574:
541:
457:
397:
368:
330:
295:
266:
216:
164:
67:algebraic geometry
63:algebraic topology
47:
1615:Monodromy theorem
1371:
1341:Analogous to the
1254:in this context.
1244:permutation group
1226:fundamental group
1037:fundamental group
722:analytic function
600:
571:
528:
513:
454:
413:fundamental group
394:
356:
321:
292:
257:
155:
127:topological space
123:locally connected
41:(an example of a
27:complex logarithm
1805:
1783:Complex analysis
1745:
1718:
1716:
1691:
1671:
1665:
1664:
1662:
1660:
1637:
1610:Monodromy matrix
1498:in the variable
1472:
1470:
1469:
1464:
1462:
1461:
1444:
1442:
1441:
1436:
1431:
1430:
1389:
1387:
1386:
1381:
1373:
1372:
1364:
1263:principal bundle
1230:
1124:
1122:
1121:
1116:
1108:
1107:
1089:
1088:
1010:
992:
990:
989:
984:
948:
934:
932:
931:
926:
918:
910:
892:
890:
889:
884:
882:
851:
775:
771:
769:
768:
763:
749:
737:
733:
714:complex analysis
699:
672:
659:
651:
624:
615:monodromy action
612:
610:
609:
604:
602:
601:
593:
583:
581:
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572:
564:
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542:
540:
536:
535:
531:
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447:
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429:
406:
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403:
398:
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387:
377:
375:
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369:
358:
357:
349:
339:
337:
336:
331:
323:
322:
314:
304:
302:
301:
296:
294:
293:
285:
275:
273:
272:
267:
259:
258:
250:
236:
232:
225:
223:
222:
217:
206:
205:
173:
171:
170:
165:
157:
156:
148:
132:
129:with base point
120:
1813:
1812:
1808:
1807:
1806:
1804:
1803:
1802:
1798:Homotopy theory
1768:
1767:
1727:
1722:
1721:
1672:
1668:
1658:
1656:
1654:
1638:
1634:
1629:
1601:
1591:leading to the
1578:Riemann surface
1569:In the case of
1542:field extension
1515:polynomial ring
1509:, which is the
1484:
1457:
1456:
1454:
1451:
1450:
1426:
1425:
1414:
1411:
1410:
1363:
1362:
1354:
1351:
1350:
1349:of a fibration
1331:
1291:is to define a
1270:smooth manifold
1252:monodromy group
1233:
1228:
1170:
1154:Vladimir Kostov
1142:
1097:
1093:
1084:
1080:
1078:
1075:
1074:
1063:
1017:
1005:
998:infinite cyclic
957:
954:
953:
939:
914:
906:
904:
901:
900:
880:
879:
847:
831:
825:
824:
802:
786:
784:
781:
780:
773:
745:
743:
740:
739:
735:
724:
710:
696:
689:
675:
670:
669:
666:monodromy group
657:
648:
642:
627:
622:
621:
592:
591:
589:
586:
585:
563:
562:
560:
557:
556:
552:
520:
519:
505:
504:
503:
499:
493:
489:
488:
484:
478:
474:
472:
469:
468:
446:
445:
443:
440:
439:
434:, and that the
431:
419:
415:
386:
385:
383:
380:
379:
348:
347:
345:
342:
341:
313:
312:
310:
307:
306:
284:
283:
281:
278:
277:
249:
248:
246:
243:
242:
234:
227:
198:
194:
186:
183:
182:
147:
146:
138:
135:
134:
130:
118:
115:
99:monodromy group
43:Riemann surface
19:
12:
11:
5:
1811:
1801:
1800:
1795:
1790:
1785:
1780:
1766:
1765:
1762:
1756:
1749:
1746:
1726:
1723:
1720:
1719:
1666:
1652:
1631:
1630:
1628:
1625:
1624:
1623:
1617:
1612:
1607:
1600:
1597:
1549:Galois closure
1483:
1480:
1460:
1434:
1429:
1424:
1421:
1418:
1379:
1376:
1370:
1367:
1361:
1358:
1330:
1327:
1231:
1186:path-connected
1184:(we assume it
1176:, and use the
1169:
1166:
1150:Carlos Simpson
1146:Pierre Deligne
1140:
1114:
1111:
1106:
1103:
1100:
1096:
1092:
1087:
1083:
1061:
1016:
1013:
994:
993:
982:
979:
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967:
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917:
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837:
834:
832:
830:
827:
826:
823:
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811:
808:
805:
803:
801:
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792:
789:
788:
761:
758:
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748:
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694:
687:
673:
646:
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599:
596:
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567:
539:
534:
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509:
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477:
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417:
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361:
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329:
326:
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215:
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114:
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17:
9:
6:
4:
3:
2:
1810:
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1789:
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1784:
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1715:
1711:
1707:
1703:
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1695:
1690:
1685:
1682:(1): 83ā108,
1681:
1677:
1670:
1655:
1653:9783658106195
1649:
1645:
1644:
1636:
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1621:
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1611:
1608:
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1602:
1596:
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1573: =
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1567:
1565:
1561:
1557:
1553:
1550:
1545:
1543:
1539:
1535:
1531:
1527:
1523:
1520:. An element
1519:
1516:
1512:
1508:
1505:
1501:
1497:
1493:
1489:
1479:
1477:
1448:
1422:
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1397:
1393:
1377:
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1321: Ć
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1312:
1308:
1304:
1300:
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1295:
1290:
1286:
1282:
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1274:
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1267:
1264:
1260:
1255:
1253:
1249:
1245:
1241:
1237:
1227:
1223:
1220: ā
1219:
1215:
1211:
1207:
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1199:
1195:
1191:
1187:
1183:
1179:
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1165:
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1159:
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1128:
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1101:
1098:
1094:
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1072:
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1022:
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1008:
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999:
980:
977:
974:
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950:
946:
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919:
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899:
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844:
841:
835:
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828:
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723:
719:
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663:
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649:
639:
635:
631:
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594:
565:
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532:
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516:
507:
500:
494:
490:
485:
479:
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448:
437:
427:
423:
414:
410:
388:
362:
350:
340:the endpoint
327:
324:
315:
286:
263:
260:
251:
240:
231:
226:. For a loop
210:
202:
199:
195:
191:
188:
181:
177:
161:
149:
143:
140:
128:
124:
110:
108:
104:
100:
96:
95:single-valued
92:
88:
84:
83:covering maps
80:
76:
72:
68:
64:
60:
56:
52:
44:
40:
36:
32:
28:
23:
16:
1736:
1689:math/0206298
1679:
1675:
1669:
1657:. Retrieved
1642:
1635:
1586:
1574:
1570:
1568:
1563:
1560:Galois group
1555:
1551:
1546:
1537:
1533:
1529:
1525:
1521:
1517:
1506:
1499:
1491:
1487:
1485:
1446:
1409:: Consider
1404:
1399:
1395:
1346:
1340:
1322:
1318:
1314:
1310:
1306:
1302:
1298:
1292:
1288:
1284:
1280:
1272:
1265:
1256:
1251:
1247:
1239:
1235:
1221:
1217:
1213:
1212:again above
1209:
1205:
1201:
1197:
1193:
1189:
1181:
1171:
1161:
1157:
1138:
1134:
1130:
1070:
1066:
1059:
1057:
1044:
1040:
1028:
1018:
1006:
995:
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940:
937:
895:
729:
725:
711:
701:
692:
685:
681:
677:
665:
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629:
619:homomorphism
614:
425:
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409:group action
229:
116:
106:
98:
94:
87:ramification
78:
54:
48:
34:
30:
15:
1760:TAC Reprint
1733:"Monodromy"
1605:Braid group
1204:above
1164:) as well.
467:is exactly
237:, denote a
75:singularity
51:mathematics
1772:Categories
1725:References
1676:J. Algebra
1407:foliations
1277:connection
436:stabilizer
133:, and let
113:Definition
1751:R. Brown
1743:EMS Press
1714:119634752
1659:5 October
1502:over the
1375:→
1369:~
1174:fibration
1091:⋯
978:π
859:
853:∣
845:∈
813:
751:∖
652:into the
598:~
584:based at
569:~
526:~
511:~
491:π
480:∗
452:~
392:~
354:~
351:γ
328:γ
325:⋅
319:~
290:~
287:γ
261:∈
255:~
233:based at
200:−
159:→
153:~
107:polydromy
91:functions
79:monodromy
55:monodromy
1599:See also
1392:groupoid
1294:holonomy
1002:helicoid
176:covering
39:helicoid
1755:(2006).
1706:2091962
1513:of the
1268:over a
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680:,
660:is the
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1009:> 0
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1684:arXiv
1627:Notes
1532:) of
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276:, by
180:fiber
178:with
174:be a
103:group
1661:2017
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1148:and
949:but
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239:lift
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